Required arm length = 37.7974 +/- 0.02 [m]
This is a preliminary result of the estimation of the Arm length tolerance.
This number was obtained from a simulation based on Optickle.
Note that the simulation was done by considering misplacements in only the arm lengths while keeping PRCL, SRCL and MICH at the ideal lengths.
Therefore the tolerance will be somewhat tighter if misplacements in the central part are taken into account.
Next : check 3f signals, and include misplacements in PRCL, SRCL and MICH.
(Background)
We will re-position the ETMY/Y suspensions to adjust the arm lenghts during the coming vent.
To get a reasonable sensing matrix for LSC, the arm length must be adjisted within a certain precision.
So we need to know the tolerance of the arm lengths.
(How to estimate)
Optickle, a frequency domiain interferomtere simulator, is used to model the response of the 40m interferometer.
I buit a 40m model in Optickle, and in this model every optical distance is adjusted to the perfect length.
Then some offsets are added on the macroscopic position of ETMs to see what will happen in the LSC sensing matrix.
When putting the offsets, the amount of offsets are randomly assigned with a Gaussian distribution (see Figure.1).
Therefore the calculation is a Monte-Calro style, but this doesn't have to be a Monte-Calro
because the parameter space is only 2-dimensions (i.e. X-arm and Y-arm length) and it can be done by simply scanning the 2-dimentional parameter space.
The reason why the Monte-Calro style was chosen is because I wanted to expand this simulation
to a more general simulation which can handle PRCL, SRCL and MICH misplacements as well.
This time I ran the Monte-Calro 1000 times.
Figure.1 History of random walk in X-Y arm lengths parameter space.
The position of ETMY and ETMX are randomly chosen with a Gaussian distribution function in every simulation.
This example was generated when \sigma_x = \sigma_y = 2 cm, where \sigma is the standard deviation of
the Gaussian function. The number of simulation is 1000 times.
(Criteria)
I made two criteria for the acceptable sensing matrix as follows :
(1) The decrease in the optical gain of the important signals (diagonal signals) must be within a factor of 3 (factor of ~ 0.5 in log scale).
(2) MICH and SRCL signals are separated within a range of 60 - 120 deg in their demodulation phases on POP55.
(Results1 : sensing matrix)
Figure.2 shows the resultant sensing matrix with histograms when \sigma_x = \sigma_y = 2,
where \sigma_x, \sigma_y are the given standard deviation in the position of ETMX and ETMY.
The diagonal signals (in red-rectangular window) shows variation in their optical gain within a factor of 0.5 in log scale (factor of 3 in linear scale).
This satisfies my requirement (1) mentioned in the last section.
Figure.2 A sensing matrix of the 40-m DRFPMI while changing the position of ETMX/Y by \sigma = 2 cm. For convenience, only REFL11, AS55, POP11 and POP55 are shown. They are the designed signal ports that
mentioned in the aLIGO LSC document ( T1000298). In all the histograms, x-axis represents the optical gain in log scale in units of [W/m]. The y-axis is the number of events. The diagonal ports are surrounded by red rectangular window.
(Results2 : demodulation phase of MICH and SRCL on POP55)
Now a special attention should be payed on the MICH and SRCL signals on POP55.
Since MICH and SRCL are designed to be taken from POP55, they must be nicely separated in their demodulation phases.
Therefore the demodulation phase of MICH and SRCL has to be carefully examined.
The plot in Figure.3 is the resultant phase difference between MICH and SRCL on POP55 when \sigma_x = \sigma_y = 2 cm.
As shown in the plot the phase are always within a range of 60 - 120 deg, which satisfies my requirement (2) mentioned in the last section.
Figure.3 Difference in the demodulation phase of MICH and SRCL on POP55.
x-axis is the difference in the demodulation phase of MICH and SRCL, and y-axis the number of events.
(Notes on the Optickle model)
Optickle that I used is the one downloaded from the MIT CVS server and I believe this is the latest version.
In my current simulation I omitted some foldng mirrors including PR3, SR2 and SR3.
If those mirrors are added on the model, loss from those mirrors will affect the build up powers in all the cavities and hence changes the sensinag matrix somewhat.
I assumed that each optic has loss of 50 ppm in its HR surface.
Input power, after the MC, of 1 W is assumed.
The modulation depth are all 0.1 rad for 11MHz and 55MHz.
The model files were uploaded on the MIT CVS server and files reside under /export/cvs/iscmodeling/40m/fullIFO_Optickle.
More information about the CVS server can be found on aLIGO wiki.
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